Inverse Trig Functions

It was horribly boring teaching how to calculate indefinite integrals which were inverse trig functions so I mixed up in the last class. I didn't give them any of the formulas. Instead I had them calculate some derivatives and rewrite as anti--derivatives. It turned out fairly well. Here is the pdf I gave to the class. I had them work in groups of 3 to 5. I walked around listening to them talk and would randomly jump in when I heard something interesting.

Observation

I was observed by the Principle. It went... okay.

You didn't teach this...

Last class we introduced the binomial theorem. My approach was to have them expand \sum_{k=0}^{n} n_C_k x^n y^{n-k} without them knowing that. I explained it to them this way: your job is to identify n (the power to which the binomial is being raise). Example: (3a -2b)^4. Start by seeing that n=4 and start writing 4_C_0 (3a)^4(-2b)^0 + 4_C_1 (3a)^3(-b^1) + ... This was prefaced by a discussion about ascending and descending terms so they would know what it meant for the 3a terms to descend and for the -2b terms to descend. That was prefaced by expanding (a+b)^n for n=0,1,2,3 to see that ascending and descending terms naturally occur in binomial expansions.

The next day I assign two problems for a warm-up:
(1) (2a - 3b)^4
(2) (5x + y^2)^5

Most students make the error of expanding (1) without the correct parentheses around the 2a and -3b terms which I correct when they come to show me their work. They go on to successfully expand (2).

Several of my better students do (1) correctly but complain that I had never "taught" them how to do (2).

Pointcare's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles

I enjoyed the book very much because of its thorough account of the major and eccentric people who worked to prove or disprove the Poincare Conjecture. I was either working on my undergraduate or in grad school when the Perelman papers were published. I remember hearing about it from faculty and fellow graduate students but that is not why I am mentioning it here.

The descriptions of the amount of (sometimes "wasted") effort on solving the conjecture illustrate the difficulty of the problem and the tenacity needed to solve ANY problem. My students often seem to think they are either good or bad at math. Either opinion is reason to give up if the solution doesn't come quickly. I don't know how I will do this but I want to find a couple excerpts about how either people kept returning to the problem or were very occupied with it. I think a major point to discuss would be that a problem has an essential question to be answered (which will hopefully tell us something interesting and novel about the world) which is different from what is learned by working out the technical details. I would hope this would turn into a discussion about why we learn math which hopefully is because we have insatiable curiosities.

Today was a better day...

My pre-calculus class with 75% seniors was better today. It was on the binomial theorem and Pascal's triangle. I started with having them expand (a+b)^n for n=0,1,2,3 by hand. We then isolated the coefficients, found a pattern and soon had Pascal's triangle. We then segued into naming terms on the triangle by using n and r to stand for the line and term on the line where we start counting from 0 (my explanation was that the first line came from expanding from n=0 so we start from zero). We then moved to the choice function. Give your calculator n and r and it spits out the term you want! Go back to the expansion talking about how each of the terms ascends or descends and how the powers always add up to n (the power to which you rose the binomial).

Now we put it together. Want to expand a binomial? Let the power be n, set up the ascending and descending terms, find the coefficients with the nCr and simplify. Go over how subtracting in the binomial changes it (alternating signs). Go over how to deal with coefficients in the binomial other than 1 (don't forget to raise the whole monomial to a power).

Geometry had a test. Lots of kids still don't understand the idea of an exact answer. I don't blame them but I am tired of hearing, "what is the real answer? I have tan36." I understand to them math is numbers but I have not figured out how to convey that those are numbers...

My Daily Routine

If I take public transportation:

Wake at 5am
Catch the 5:30am train
Arrive on campus at about 5:50-6:00am, prepare by writing lectures, preparing notes, making quizzes and tests
7:45am - 2:30pm teach with one 1:20 free time
2:30-4:30pm coach
5pm exercise (hopefully for an hour)
6:30pm or after, get home help with family chores, feed kids, cook, etc.
9pm (hopefully) kids are laying down, light heartedly work on something
midnight usually go to sleep or sleeping

A hopeful start with a frustrating end

I taught two classes today. Honors Geometry and Pre-Calculus. The first class went very well. We had a review session. Several students either started or completed work on extra credit assignments. It was good.

The last period of the day arrived. It started out well enough. Students worked on review problems from the previous class. I introduced new material then everyone just kept talking and talking and talking. Half the class is seniors. I'll just stop here. This has been cathartic enough.